What is Irrational: Definition and 350 Discussions

Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. The term is used, usually pejoratively, to describe thinking and actions that are, or appear to be, less useful, or more illogical than other more rational alternatives.Irrational behaviors of individuals include taking offense or becoming angry about a situation that has not yet occurred, expressing emotions exaggeratedly (such as crying hysterically), maintaining unrealistic expectations, engaging in irresponsible conduct such as problem intoxication, disorganization, and falling victim to confidence tricks. People with a mental illness like schizophrenia may exhibit irrational paranoia.
These more contemporary normative conceptions of what constitutes a manifestation of irrationality are difficult to demonstrate empirically because it is not clear by whose standards we are to judge the behavior rational or irrational.

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  1. J

    Converting Equations to Binary & Irrational Numbers

    ... can we convert this equation to binary notation? Also another one, would an irrational number be irrational in any number format?
  2. S

    Is the equation 4x^(2) + 5y^(2) = 2 solvable with rational roots?

    Show that the equation 4x^(2) + 5y^(2) = 2 has no rational solutions. Can this be done graphically?
  3. P

    *Proof* Sum of Rational and Irrational Numbers

    Homework Statement Prove by contradiction: If a and b are rational numbers and b != 0, and r is an irrational number, then a+br is irrational. In addition, I am to use only properties of integers, the definitions of rational and irrational numbers, and algebra. You guys should also know that...
  4. M

    Function continuous in exactly the irrational points

    Give an example of a function f:(0,1)-->Reals which is continuous at exactly the irrational points in (0,1). I think the function f such that f(x)=1/n if x is rational in (0,1) (x=m/n for some n not 0) and f(x)= 0 if x is irrational in (0,1) should work. I get the reason why f is continuous...
  5. M

    Pigeonhole Principle & irrational numbers

    Homework Statement Let x be an irrational number. Show that the absolute value of the difference between jx and the nearest integer to jx is less than 1/n for some positive integer j not exceeding n. Homework Equations The Attempt at a Solution Ok, I know that it should be solved using...
  6. Q

    Prove there is an irrational between any two rationals

    Homework Statement Prove that there exists an irrational between any two rationals. Homework Equations The Attempt at a Solution How would one do this? So far I've proven there is an irrational between any rational and irrational, any irrational and rational, that there's a...
  7. E

    Proove that the cubic root of 2 + the square root of 2 is irrational

    How do you show that the cubic root of two + the square root of two is irrational? I can easily show that each of these numbers is irrational, but not the sum :/.
  8. N

    Between a rational and a irrational

    Hello. Between a rational and a irrational is there a rational? and a irrational? and vice-versa? I know that between 2 rationals there is a rational and a irrational and that between 2 irrationals there is a rational and a irrational, but i cannot figure this out... please help. Thanks.
  9. K

    Is x Irrational if x² Is Irrational?

    Homework Statement Prove that if x^2 is irrational then x must be irrational. Homework Equations The Attempt at a Solution Maybe do proof by contradiction. I'm not really sure where to start.
  10. H

    Proving an irrational to an irrational is rational

    Homework Statement prove that it is possible that an irrational number raised to another irrational, can be rational. you are given root2 to root2 to root2 Homework Equations The Attempt at a Solution i have shown that root2 to root2 to root2 is rational, but would appreciate a...
  11. X

    Irrational Number Approximation Error Explained

    Question: Using the fact that \sqrt{2} is irrational, we can actually come up with some interesting facts about other numbers. Consider the number t=1/\sqrt{2}, which is also irrational. Let a and b be positive integers, and a<b. We will prove that any rational approximation a/b of t will...
  12. A

    I wih prove (sqrt(2))^3 is irrational number

    hi well I'm having truple in proving this (sqrt(2))^3 is irrational number!
  13. A

    Definition & Properties of Irrational Numbers

    How do we exactly define irrational numbers.. ive asked this before... but id like to know about any infinite series, if any which is used to define irrational numbers... and how can one prove properties of basic operations for irrational numbers Thanks
  14. M

    Integrating irrational functions

    Hello everyone! I was wondering.. if you could help me calculate some integrals: It's not for Homework or something, just my curiosity: \displaystyle{\int}\sqrt[3]{x^2-1} dx What would you suggest? I tried substitution, thou it seems to me useless. Are these integrals common in...
  15. E

    Is it possible to find function that describes irrational things?

    i.e. decimals of an irrational number, pi(n), etc. The function itself, f(n), can change with n, as long as it changes in a patterned way. This seems like a straight foreword question, either you can for all, or you can't for all. A simple example of a changing function would be the...
  16. K

    Is i Rational or Irrational? Decoding the Nature of Imaginary Numbers

    I've been thinkng about this one for a while. Is i rational or irrational. i is an imaginary number, so logically, it would be irrational. But \frac{-1}{i} = i so it has a fractional equivilant. But then, it doesn't have a real number decimal equivilant... So, what is it? Is i rational or...
  17. J

    Is there always at least one irrational number between any two rational numbers?

    and, consequently, infinitely many. I am new to proofs so could you please check if this proof is correct? Let x be an irrational number in the interval In = [an, bn], where an and bn are both rational numbers, in the form p/q. Let z be the distance between x and an, So: x - an = x...
  18. R

    Radioactive materials irrational fears

    I know this will sound nuts, because it kind of is, but I though maybe someone could talk some sense into me. I suffer from ocd and irrational fears about radioactive substances. I realize that we are surrounded by and constantly bombarded by radiation. That doesn't bother me. Nor does having...
  19. F

    Prove 7th root of 7 is irrational

    1st I assume it is rational so: 7^(1/7) = m/n then 7 = (m^7)/(n^7) implies m^7 is a multiple of 7. Means m^7 = 0 mod 7 So, using fermats little theorem.. m^7 = m mod 7 for m to be in the class of 0 it has to be a multiple of 7. Now set m = 7k, so 7n^7 = 49k^7 But...
  20. P

    Proving Integral of an Irrational Function

    Hey, Homework Statement (From an Integration Table) Prove, {\int}{\sqrt {{{a}^{2}} - {{x}^{2}}}}{dx} = {{\frac {1}{2}}{\left(}{{{x}{\sqrt {{{a}^{2}} - {{x}^{2}}}}} + {{{a}^{2}}{\arcsin{\frac {x}{a}}}}}{\right)}}{,}{\,}{\,}{\,}{\,}{\,}{\,}{{{|}{x}{|}}{\leq}{{|}{a}{|}}} Homework...
  21. E

    Continuous at irrational points

    [SOLVED] continuous at irrational points Homework Statement Every rational x can be written in the form m/n where n>0, and m and n are integers without any common divisors. When x = 0, we take n=1. Consider the function f defined on the reals by f(x) = 0 if x is irrational and f(x) = 1/n if x...
  22. V

    What is the difference between rational and irrational numbers?

    Can someone pls help me on "rational and Irrational numbers". Esp. on Decimals. I can't classify if it is rational or irrational.
  23. X

    Prove that cos 20 is irrational

    I've started to work on the it, just tell me if I'm on the right track. cos (45-30) = (\sqrt{3} + 1) / 2\sqrt{2} so cos 15 is irrational. cos3x = 4cos^3x - 3cos x \Rightarrow cos 5 is irrational cos 4x... cos 20 If this is a bad way, maybe someone knows a better one. Here is what I think...
  24. S

    Can statistics support irrational thinking?

    Here is the scenario. Picture three people standing at a playground watching the children play. The first is grouchy and sees all the busy activity as stressful noise. The second person is in a good mood and sees the child play as relaxing and fun. The third person is full of anxiety and sees...
  25. A

    Proof that sqrt(6)-sqrt(2)-sqrt(3) is irrational

    I want to prove that \sqrt 6 - \sqrt 2- \sqrt 3 is irrational. I already know that \sqrt 2+\sqrt 3 is irrational (by squaring it). I would like a proof that doesn't use a polynomial and the rational root theorem. Thanks.
  26. S

    About irrational number

    wat are the answers for these in terms of rational,irrational (irrational number)*(any +ve integer) = ? (any +ve integer) - (irrational number)*(any integer) = ? are the answers also irrational numbers
  27. K

    Proving Irrationality of 2√2, 2-√2, 17√(1/2)

    I know that √2 is irrational (and I've seen the proof). Now, what is the fastest way to justify that 2√2, 2-√2, 17√(1/2) are irrational? (they definitely "seem" to be irrational numbers to me) Can all/any these follow immediately from the fact that √2 is irrational? Thanks!
  28. G

    Prove that the cuberoot of 2 is irrational

    [SOLVED] Prove that the cuberoot of 2 is irrational Homework Statement Prove that the cuberoot of 2 is irrational The Attempt at a Solution Assume it is rational, and find a contradiction. 2^(1/3) = a/b, where a, b are integers, where a/b is in lowest terms, and where b != 0. 2 = a^3 /...
  29. D

    Add Two Irrational Surds to Get Another Surd

    Hi, does somebody know an example of two surds that, added together, give another surd? By 'surd' I mean here 'irrational surd', as opposed to \sqrt 4 + \sqrt 9 = \sqrt 25.
  30. R

    Proving Real Number Limit with Irrational Sequence"</code>

    Homework Statement Prove, for every L which is in the real number system, there exists a sequence (qn)which is a proper subset of the irrationals such that the limit as n approaches infinity of qn=L
  31. R

    Someone help me understand how e is irrational proof

    http://web01.shu.edu/projects/reals/infinity/irrat_nm.html Well there is the proof i am reading and trying to understand... can someone tell me how they knew that 0<R_n<\frac{3}{(n+1)!}
  32. P

    Proof that Log2 of 5 is irrational

    Homework Statement Prove that log2 of 5 is irrational. Homework Equations None. The Attempt at a Solution I just had a glimpse of the actual solution, but I'm wondering if mine would work too. 2^(a/b) = 5 square both sides... 2^(2a/b) =25 2 = 25^(b/2a) (b/2a) = log25 of 2 b =...
  33. S

    Exploring Limits with Irrational Exponents

    limit proof?? well what i am trying to understand,actually proof is if we can get with the limit inside a power (exponent) if the exponent is irrational. Say we have any sequence (a_n) or any function f(x), let p be irrational then can we do the following, if yes why, if not why? 1...
  34. C

    Irrational polynomial equation

    what's the polynomial equation which sqrt2 + sqrt3 satisfies ?
  35. H

    Proving Irrationality of Sums and Products of Irrational Numbers

    hi i m hashim i want to solve a qquestion 1.if x is rational & y is irrational proof x+y is irrational? 2. if x not equal to zero...y irrational proof x\y is irrational?? 3.if x,y is irrational ..dose it implise to x+y is irrational or x*y is irrational thanks please hashim
  36. A

    Prove the Irrationality of the Golden Ratio & Phi

    The golden ratio is irrational. Do you know any clever proofs for this fact? I put this here, because it's not homework--only more of a discussion.
  37. A

    Finding the unknowns of irrational equation

    Homework Statement For the following irrational equation x^2 + 7x + 10 + \sqrt{x^2 + 7x + 12} = 0 Find all possible unknown of X. Homework Equations None. Just your ability to solve equations. The Attempt at a Solution First of all, I am not allowed to use a calculator to...
  38. T

    Proving that cube root 7 is irrational

    Hi guys, How would you prove that \sqrt[3]{7} is irrational without using the unique factorization thrm? I tried proving that \sqrt[3]{7} is rational but it didn't seem to get me anywhere... Thanks EDIT: Looks like I posted this in the wrong forum.
  39. S

    Analysis - irrational number or positive integer

    Homework Statement Let n and k be positive integers. Show that k^{1/n} is either a positive integer or an irrational number. The Attempt at a Solution I set q = k^{1/n}. Then I set q = \frac{m}{p} . (Where m and p don't have common factors.) Then m^n = k * p^n . So then k is a factor...
  40. S

    Analysis Question - irrational and rational numbers - proof

    This isn't really a question about homework specifically, it's more just that I don't understand part of my chapter...I am just starting Principles of Mathematical Analysis by Ruben... Here is what I don't understand: It is proving that p^2 = 2 is not satisfied by any rational p. And it...
  41. L

    Rational and Irrational Number Set proof.

    Hello, here is my problem: how can i prove that if a\in\mathbf{Q} and t\in\mathbf{I}, then a+t\in\mathbf{I} and at\in\mathbf{I}? My original thought was to show that neither a+t or at can be belong to N, Z, or Q, thus they must belong to I. However I'm not certain if that train of thought...
  42. P

    A quick question on Irrational powers

    I wish to prove that for f(x)=x^x, its domain is: {x E R, x > 0}U{xEZ,x<0}. I reevaluated to e^(xlnx), obviously that did not help. Is there an algorithm/formula/something that can evaluate irrational powers, so that it can help me with this?
  43. L

    Irrational or Rational [Newton-Raphson]

    Homework Statement How are you able to determine if a solution is rational or irrational Homework Equations - The Attempt at a Solution - :confused: I'm pretty sure it will be something basic I'm forgetting, thanks for any help.
  44. M

    Irrational natural log integral

    Homework Statement the indefinite integral of (1+lnx)^(1/2)/(xlnx) dx Homework Equations n/a The Attempt at a Solution There aren't any x^2 in the root sign, so I don't think it can be a trig substitution. The only logical u sub I see is to let u=lnx. In that case, du=dx/x so the...
  45. P

    Is Honor Still Relevant in Online Forums?

    What is honor good for on internet forums?
  46. A

    Difference of two irrational numbers

    Im wondering if its possible given x,y irrational, that x-y is rational (other than the case x=y). The reason I am asking this is that I am reading a book on measure theory and they try to construct a non measurable set and they start with an equivalence relation on [0,1} x~y if x-y is rational...
  47. JasonRox

    Proving something is irrational.

    Homework Statement Prove 5^{1/3} - 3^{1/4} is irrational. Homework Equations http://www.purplemath.com/modules/solvpoly.htm The Attempt at a Solution Ok, what I have tried doing is using the about Rational Roots property by letting x = 5^{1/3} - 3^{1/4} and trying to pull out a...
  48. F

    Randomness of digits of irrational numbers.

    How random are the digits of irrational numbers? Can it be said of them (i.e. pi=3.14159...) that given any arbitrarily long string of digits it must occur at some point in any irrational number? And would anyone know of anywhere I could find out more on this topic?
  49. murshid_islam

    Meaning of irrational exponent

    i know what the meaning of a^p is when p is an integer or rational. e.g., a^3 = a.a.a or a^{\frac{1}{5}} is such a number that when multiplied five times gives the number a. but what is the menaing of a^p when p is an irrational number?
  50. Pythagorean

    The Might of Occam's Blade Stops Irrational Numbers

    I have discovered the mighty blade of Occam, I shall destroy all who advocate the existence of irratonal numbers. Spiders are your gods.
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